Optimal. Leaf size=161 \[ -\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {16}{35} c^3 x \text {ArcTan}(a x)+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \text {ArcTan}(a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)-\frac {8 c^3 \log \left (1+a^2 x^2\right )}{35 a} \]
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Rubi [A]
time = 0.05, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4998, 4930,
266} \begin {gather*} \frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \text {ArcTan}(a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}-\frac {3 c^3 \left (a^2 x^2+1\right )^2}{70 a}-\frac {4 c^3 \left (a^2 x^2+1\right )}{35 a}-\frac {8 c^3 \log \left (a^2 x^2+1\right )}{35 a}+\frac {16}{35} c^3 x \text {ArcTan}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4930
Rule 4998
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx\\ &=-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx\\ &=-\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac {1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)-\frac {1}{35} \left (16 a c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)-\frac {8 c^3 \log \left (1+a^2 x^2\right )}{35 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 83, normalized size = 0.52 \begin {gather*} \frac {c^3 \left (-a^2 x^2 \left (57+24 a^2 x^2+5 a^4 x^4\right )+6 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \text {ArcTan}(a x)-48 \log \left (1+a^2 x^2\right )\right )}{210 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 102, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \left (\frac {5 a^{6} x^{6}}{6}+4 a^{4} x^{4}+\frac {19 a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(102\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \left (\frac {5 a^{6} x^{6}}{6}+4 a^{4} x^{4}+\frac {19 a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(102\) |
risch | \(-\frac {i c^{3} x \left (5 a^{6} x^{6}+21 a^{4} x^{4}+35 a^{2} x^{2}+35\right ) \ln \left (i a x +1\right )}{70}+\frac {i c^{3} a^{6} x^{7} \ln \left (-i a x +1\right )}{14}-\frac {a^{5} c^{3} x^{6}}{42}+\frac {3 i c^{3} a^{4} x^{5} \ln \left (-i a x +1\right )}{10}-\frac {4 a^{3} c^{3} x^{4}}{35}+\frac {i c^{3} a^{2} x^{3} \ln \left (-i a x +1\right )}{2}-\frac {19 a \,c^{3} x^{2}}{70}+\frac {i c^{3} x \ln \left (-i a x +1\right )}{2}-\frac {8 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{35 a}\) | \(168\) |
meijerg | \(\frac {c^{3} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 a^{8} x^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{4 a}+\frac {3 c^{3} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a}+\frac {3 c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4 a}+\frac {c^{3} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 99, normalized size = 0.61 \begin {gather*} -\frac {1}{210} \, {\left (5 \, a^{4} c^{3} x^{6} + 24 \, a^{2} c^{3} x^{4} + 57 \, c^{3} x^{2} + \frac {48 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \arctan \left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.57, size = 101, normalized size = 0.63 \begin {gather*} -\frac {5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 6 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \arctan \left (a x\right )}{210 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.41, size = 117, normalized size = 0.73 \begin {gather*} \begin {cases} \frac {a^{6} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {a^{5} c^{3} x^{6}}{42} + \frac {3 a^{4} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {4 a^{3} c^{3} x^{4}}{35} + a^{2} c^{3} x^{3} \operatorname {atan}{\left (a x \right )} - \frac {19 a c^{3} x^{2}}{70} + c^{3} x \operatorname {atan}{\left (a x \right )} - \frac {8 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{35 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 89, normalized size = 0.55 \begin {gather*} -\frac {c^3\,\left (48\,\ln \left (a^2\,x^2+1\right )+57\,a^2\,x^2+24\,a^4\,x^4+5\,a^6\,x^6-210\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-126\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )-30\,a^7\,x^7\,\mathrm {atan}\left (a\,x\right )-210\,a\,x\,\mathrm {atan}\left (a\,x\right )\right )}{210\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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